Eigenvalue analysis generalized mass 3

[vanstoja]

My understanding of the meaning of generalized mass from an eigenvalue analysis is that it represents inverse damping, ie, the higher the generalized mass, the lower the damping at the calculated modal frequency. A supplier provided eigenvalue calculations for a 100+ node pump and motor giving generalized masses for 100+ "resonant" frequencies ranging from less than 1 to over 4000. Are there any general guidelines for lumping the GM numbers into a few categories between say "overdamped" and "zero damping"?
vanstoja

 

[MikeyP]

An eigenvalue analysis obtained from FE software will usually have nothing to do with damping. It involves only the mass and stiffness matrices. A special form of linear system eigenvalue analysis known as "complex mode analysis" does incorporate a damping matrix, but I know of no commercial FE package which does this. In fact FE pacakages are notoriously poor for their representation of damping. The vast majority only allow simple viscous damping models, usually mass proportional damping or mass and stiffness proportional (Rayleigh) damping, to be included in a dynamic model. These damping models often bear little resemblence to any real world damping levels.

"Generalised mass" is a term which usually means a representation of the "physical" mass matrix which has been transformed from "physical" (also called "nodal" in FE) coordinates to generalised (also called "modal&quot coordinates.

The eigenvalue analysis produces eigenvalues, which are related to the system natural frequencies (actually 2*pi*f^2), and eigenvectors, which are the mode shapes associated with each of those frequencies. If all these eigenvectors put together into one matrix, with each column of the matrix being one eigenvector, then we have the "modal matrix" of the system. Each column may be scaled in any way you wish. Often the columns are scaled so that each has a maximum value of 1.

The physical mass, stiffness and (if needed) damping matrices can be converted to generalised coordinates by post-multiplying by the modal matrix and pre-multiplying by the transpose of the modal matrix.

These new generalised mass and stiffness matrices will always be diagonal. The same is true of the damping matrix provided a Rayleigh damping model has been assumed. There is now a generalised mass, stiffness and damping value associated with each mode. However, these generalised values are not unique. They depend entirely on the scaling of the eigenvectors in the modal matrix. A generalised mass matrix is pretty useless unless you know the modal matrix which was used to calculate it.

One form of modal scaling which is often used is "unity modal mass". The modal matrix is scaled such that the generalised mass matrix is the identity matrix.

I hope this sheds some light on your problem, but I must stress again that generalised mass values have nothing to do with damping.

Michael

 

[MikeyP]

Correction: eigenvalues should be related to frequency by

(2*pi*f)^2

not

2*pi*f^2

M

 

[PVI]

If you do want to introduce damping effects you need to perform complex modal analysis, ie Stiffness matrix may have a real part and a complex part to introduce frequency dependent materials properties. By doing this, you do not need to introduce numerical damping to tune or correlate your FE model to physical one.

Philippe

 

[vanstoja]

Thanks for the prompt and enlightening replies...but my "understanding" of the generalized mass-damping connection actually comes from the original Shock and Vibration Handbook (S&VH)(1961)Chapt. 48 on "Vibration Induced by Acoustic Waves" by Hubbard & Houbolt of NASA. In a subsection, "Inclusion of Structural and Air Damping", generalized mass, B_n is the acceleration coefficient in the response equation 48.15 and further on a damping ratio is equated to the sum of the structure/air damping terms divided by a critical damping term defined as 2*omega_n*B_n. That seems to make generalized mass an inverse factor affecting damping ratio. This chapter didn't make the cut in the ensuing 3-volume to 1-volume change. In the 3rd (1988) edition, equation 2.93 gives a displacement equation with generalized mass in the denominator but doesn't seem to show any connection to damping ratio. Our supplier's eigenvalue analysis uses WECAN, the Westinghouse Electric answer to or version of NASA's NASTRAN. It returns mode number, frequency, generalized stiffness , generalized mass and eigenvector data that can be shown as modeshape or described qualitatively as eg. "rotor rocking - stator twisting", etc. All the frequencies are identified in one of three axial directions. Does all this make any difference to your replies? If so, I would like to rephrase the question as "How, with generalized stiffness and damping given, do I determine the relative hierarchy of the 100 or so calculated frequencies with respect to imminent disaster level if I happen to have a potent vibration source fundamental or harmonic at that frequency?" We think we've already found several prominent non-source spectral peaks that match the calculated high generalized mass frequencies. Independent analysis found the highest generalized mass frequency to be the torsional response of a big, heavy, stepped, circular cantilever beamlike structure about its rather flexible support skirt. vanstoja

 

[GregLocock]

By the way I agree with the earlier responses, modal mass has very little to do with the importance or otherwise of a mode, since it is an arbitrary scaling decision. For instance, suppose I was measuring the response of the wing mirror on a car. If I attach the shaker to the mirror then the mirror's fundamental mode will dominate the FRFs, and the modal mass referenced to the excitation point will be very small. The torsional frequency of the car's body will have a very high modal mass. If I put the shaker on the bumper at the front left of the vehicle then the modal mass of the two will be reversed, if I can even see the mirror mode. If I normalise the modal mass to give maximum response =1 then the modal mass also varies depending on where I put the shaker. Which is absurd.

"How, with generalized stiffness and damping given, do I determine the relative hierarchy of the
100 or so calculated frequencies with respect to imminent disaster level if I happen to have a
potent vibration source fundamental or harmonic at that frequency?"

That's a much better question. The answer is that you need to need to do a forced response analysis, ie drive your linear model with a representatitive forcing function and if you are a great believer in software you can even do a fatigue analysis. Concentrate on the red bits. In general you can't really do a fatigue analysis on an unknown (ie poorly understood) structure in the frequency domain - time history is the way to go, but if you have a strong tonal source you may get away with it.

However linear FE, as such, does not understand damping in any useful fashion, you will need to assign a different damping value to each mode if you wish to estimate stresses and get good correlation. A more realistic alternative is to dump your linear model into a program that does understand discrete damping in assemblies. The ones we use are called FDYNAM, FLAP and VSIGN, but I'm sure there are many alternatives, I assume Nastran, and and I know ABAQUS, has extensions to allow this. Cheers

Greg Locock